The European champions didn't finish high enough in the league to qualify for this year's competition, but victory in the final in May meant they were handed one of the Premier League's Champions League spots at the expense of Spurs.
Dawson told the The Mirror: "Of course, we are still hurting. It would've been amazing to play in the Champions League this year.
"It was hard to take for myself and for the lads who thoroughly deserved to finish in fourth place. To lose out the way we did was remarkable.
"When it goes to penalties it's one of those things – an English team against a German team, you would normally know what happens."
Tottenham are competing in this season's Europa League instead of the Champions League.